A continuum model for ion evaporation from a drop: effect of curvature and charge on ion solvation energy Original Research Article

A drop of liquid is treated as a continuum medium with surface tension γ and dielectric constant ε. The energy Δ (surface + electrostatic) required to extract a solvated ion from the drop can then be determined as part of a well posed problem as a function of the initial number z + 1 of elementary charges in the drop and its initial radius R′. Instead of a direct numerical attack on the full model, a geometrically simpler situation is analyzed, in which the drop and the solvated ion are taken to be either spheres or spherical caps after or prior to detachment, respectively. This simplified model is closely related to the full continuous problem when the radius Ri of the solvated ion is small (Ri/R′ ⪡ 1), and the main drop is not near the Rayleigh limit. This model problem is solved analytically in the limit ε ⪡ 1. When z = 0 and 1/R′ = 0 one recovers Born’s result, where Δ = 2.7 eV for monovalent ions in water, which exceeds by some 0.3 eV the experimental value for the alkali ions. In the limit of small ions one recovers the results of Gamero et al. , Δ = ΔGBorn − e2[F(z)+α]/(4πε0R′), though surface tension effects shift the constant α from 4/5 to 2/3. The effect of finite ion diameter is determined numerically for the two-spheres model. When the solvation energy at zero curvature and charge is ΔGborn, small ions do not evaporate from water drops. When this value is reduced to fit available experimental data, a narrow ion evaporation window appears for drops charged near the Rayleigh limit with z ∼ 12 or less. The domain in (z, R′) space leading to ion evaporation is broader for the case of formamide. The micro-hydrodynamic process of escape of a singly charged nanodrop from a larger drop requires a large activation energy. It is hence indistinguishable from Iribarne–Thomson ion evaporation, and radically different from a Coulomb explosion.